Quantum Class 2, Thurs 2022-09-01

W Randolph Franklin (WRF), Rensselaer Polytechnic Institute (RPI)

I edited the blog to show what we actually did.

2 Homework 2

is online, due Tues.

3 Theory vs Experiment

Sometimes the theoreticians posit something new, then the experimentalists try to build it.
1. atom bomb
2. classical computer
3. quantum computer
Other times the experimentalists (aka hackers) build something new.
1. bridges
2. airplanes
3. medieval cathedrals
4. steam engines
5. calculus
Then those may become so successful that bigger and bigger ones are built until those collapse / crash / fall down / blow up / have paradoxes.
Then the theoreticians have to come in and clean up the mess to allow more progress.
They may look for unifying themes in apparently separate problems.
Group theory in abstract (modern) algebra happened that way.
Anything discovered about groups was then valid in all the applied areas.
All theories have limits that may or may not be relevant.
Paradoxes expose limits. "All triangles are isosceles" error exposes something not covered by Euclid's axioms.
Physics Nobels tend to go to theoretical explanations of surprising experiments. Although, there is nothing in common among all Nobel winners, not even intelligence (according to Enrico Fermi).
Very rarely, experimentalists do something that the theoreticians had proven was impossible.
1. Galileo seeing sunspots.
2. Marconi radioing across Atlantic.
3. Michaelson-Morley. (However several other contemporaneous inexplicable observations all proved to have classical explanations.)
and something new is discovered / invented. Unfortunately the crackpots then seize on these rare examples...

4 Video

Quantum Computing in Under 11 Minutes daytonellwanger https://www.youtube.com/watch?v=TAzZKAdX2Tw 10:56 more technical

5 Intro to quantum computing

This is from
1. Yanofsky and Mannucci, Quantum Computing for Computer Scientists,
2. Hidary, 2nd edition, and
3. my opinions.
Isomorphism between geometry and algebra. Work in whichever domain is easier. Results carry over.
1. 2D: x+ij -> (x,y)
2. 3D: quaternions. 3D rotations are not commutative, and so quaternions are not. There's a memorial on a bridge in Trinity College Dublin where William Rowan Hamilton thought of them. Quaternions are equivalent to 3D vector algebra.
3. qbits - Bloch sphere.
qbits vs qubits? Above my pay grade.
Evidence for quanta:
1. 1905 photo-electric effect explanation. Einstein Nobel.
2. Ultraviolet catastrophe.
3. 1922 Stern Gerlach experiment. Very influential.
  
  https://en.wikipedia.org/wiki/Stern%E2%80%93Gerlach_experiment
Hidary 1.1 Quantum Computer Definition, p 3

A quantum computer is a device that leverages specific properties de- scribed by quantum mechanics to perform computation.
State vector
1. completely describes a system
2. cannot be directly observed, only measured...
3. which changes it.
Hidary 1.1 Superposition and Entanglement, p 4

2 big properties.
bra-ket notation.
can rewrite a state vector in a different basis
Hidary 1.3 The Superposition Principle, p 4
1. "The linear combination of two or more state vectors is another state vector in the same Hilbert space and describes another state of the system."
2. Generally false in classical domain.
  
  Particle can go thru only 1 of 2 slits.
  
  However vibration states in a string add.
Hidary entanglement p 7

description is confusing, defer till later.
Yanovsky p 40, vector space review
1. One qbit is a simple vector space.
2. Can change basis, e.g., to Hadamard basis. p 51.
Eigenvalues and eigenvectors, p 60
Hermetian matrices, p 63
Classical 2 slit experiment, p 86.

Probabilities add.
Quantum 2 slit experiment, p 93.

Complex waves add.

Probabilities might reduce.

6 Quantum properties - Entanglement

Crazy counterintuitive idea that's the basis for quantum speedup.
Classical metaphor for entanglement:
1. Start with a piece of paper.
2. Tear it into two halves.
3. Put each half into an envelope, seal them, and mix them up, so that you can't tell which half is in which envelope.
4. Address and mail one envelope to a friend in Australia, and the other to a friend in Greenland.
5. When the Australian opens his envelope, he knows what the Greenlander will find in his.
6. However that doesn't let the Australian send any info to the Greenlander, or vv.
7. This has been demonstrated with real qbits transported 1000 miles apart.
8. Entanglement means that if you measure one qbit then what you observe restricts what would be observed when you measure the other qbit.
9. However that does not let you communicate.
The above metaphor is inaccurate in ways that don't affect us now. (It assumes a hidden variable, which is false.)
Entanglement and superposition do funny things to operations. Consider the Controlled NOT gate. It has 2 inputs, x and y.
1. Classically, y is negated iff x=1. x doesn't change.
2. If x and y are superposed with a Hadamard basis, then y can affect x.